Numerical Methods for Differential Games: Capital Structure in an R&D Duopoly
Bibliographic record
Abstract
This paper compares two different numerical methods used to solve the same differential game. In differential games strategies of individual players are represented as continuous functions of time and are typically solutions to the optimal control problems of the players. The game is an R&D duopoly with two players: an upstream firm that is primarily engaged in research and development (the R&D firm) and whose value comes from the market valuation of these activities, and a downstream firm primarily engaged in distribution and marketing (the D&M firm). The first method is assumed to be the benchmark since it is based on discretizing the first order conditions of each player’s optimal control problem. The second method is based on making random guesses of the parameters of a second order polynomial and searching for optimal solutions. The results suggest that the second method, which is more automated and has the potential of being applied to games with higher dimensionality, can give approximate solutions to differential games similar to the one considered here. The results also provide an important theoretical outcome. They illustrate that unlike the tradeoff and pecking order models of capital structure there are many markets in which capital structure is not driven by a reversion to a target debt-to-equity ratio or a pecking order, but by maximizing firm value under strategic considerations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".